Distinction, Discretisation, and the Cost of Law: A Quantitative Sketch of Approximation Entropy

In the paper's words

The deepest question in physics is often posed as "What are the laws?", but a prior question is "What does it cost to state, and to compute with, a law?" Building on the distinction-based formalism of Spencer-Brown's Laws of Form as consolidated in the Quantum Laws of Form (QNFO) programme, and on the QNFO thesis that physical law requires only the rational numbers, we develop a quantitative sketch of approximation entropy: the irreducible information cost of translating continuous bosonic physics into discrete digital computation. We define a rational-grid discretisation of a continuous parameter, count the admissible rational states via a Farey-type estimate, and derive the scaling H_app(ε) = log₂ bigl(3/(π²ε)bigr) for the bits required to specify a parameter to absolute tolerance ε (on the unit interval [0,1] absolute and relative tolerance coincide for values of order unity). We compute a concrete value, H_app(10^-4) ≈ 11.57 bits, with full arithmetic, and project the Trotter-step cost of digital simulation of a continuous evolution, obtaining a wall N_step ∝ T²/ε. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. We compute a concrete value, $H_{\mathrm{app}}(10^{-4}) \approx 11.57$ bits, with full arithmetic, and project the Trotter-step cost of digital simulation of a continuous evolution, obtaining a wall $N_{\mathrm{step}} \propto T^{2}/\varepsilon$.

   from Abstract

2. A third, developed in v2.0 of the Two-Level Lie document, introduces approximation entropy as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, together with named structures — the Rosetta Constant, the Trotter Wall, and the Fractal Wall theorem — and a falsifiable calorimetry experimental programme [12].

   from 1. Introduction

3. We combine the two into a single figure of merit, the translation cost $C_{\mathrm{tr}}$, and exhibit one fully computed numerical example (Section 4.3).

   from 1. Introduction

4. This is directly relevant to our Section 4.3: the equilibration of a continuous quantity between two discrete subsystems, with an irreducible loss, is the physical picture behind our translation cost $C_{\mathrm{tr}}$. [5] Laws of Physics is a philosophical introduction to laws of physics, connecting them to ontology, possibility, explanation, induction, counterfactuals, time, determinism, and fundamentality, and suggesting that laws are fundamental facts that govern (the summary truncates).

   from 2. Background and Related Work

5. It supplies the philosophical frame for our opening question: whether laws are fundamental facts or constructed descriptions bears directly on whether the discretisation cost we quantify is a fact about nature or about our descriptions. [6] The Neutrino Factory ISS physics report presents the conclusions of the Physics Working Group of the international scoping study of a future Neutrino Factory and super-beam facility, carried by the international community between NuFact05 (Rome, June 21-26, 2005) and NuFact06 (Irvine, California, 24-30 August 2006).

   from 2. Background and Related Work

6. We use only the thesis stated in the title — that physical law requires only $\mathbb{Q}$ — and flag in Section 6 that this is a title-level thesis, not one we can substantiate further from the supplied material. [12] The Two-Level Lie document, v2.0, adds Part II: the Approximation Entropy, described as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, introducing the Rosetta Constant, the Trotter Wall, and the Fractal Wall theorem with a falsifiable calorimetry experimental programme.

   from 2. Background and Related Work

How it is done

1. We define a rational-grid discretisation of a continuous parameter, count the admissible rational states via a Farey-type estimate, and derive the scaling $H_{\mathrm{app}}(\varepsilon) = \log_{2}\!\bigl(3/(\pi^{2}\varepsilon)\bigr)$ for the bits required to specify a parameter to absolute tolerance $\varepsilon$ (on the unit interval $[0,1]$ absolute and relative tolerance coincide for values of order unity).

   from Abstract

2. A third, developed in v2.0 of the Two-Level Lie document, introduces approximation entropy as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, together with named structures — the Rosetta Constant, the Trotter Wall, and the Fractal Wall theorem — and a falsifiable calorimetry experimental programme [12].

   from 1. Introduction

3. Because several entries carry only truncated summaries, we confine every statement about a work to what its own supplied text states. [1] The Physics Briefing Book describes the European Particle Physics Strategy Update (EPPSU) process, which takes a bottom-up approach: the community is first invited to submit proposals (inputs) for projects it would like to see realised in the near-term, mid-term and longer-term future, with national inputs and inputs from national laboratories also important elements.

   from 2. Background and Related Work

4. We use this as the model of the size of the rational grid.

   from 3.2 Counting admissible rationals

What would falsify it

1. The framework yields a falsifiable calorimetry proposal [12] and identifies open problems.

   from Abstract

2. A third, developed in v2.0 of the Two-Level Lie document, introduces approximation entropy as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, together with named structures — the Rosetta Constant, the Trotter Wall, and the Fractal Wall theorem — and a falsifiable calorimetry experimental programme [12].

   from 1. Introduction

3. We use only the thesis stated in the title — that physical law requires only $\mathbb{Q}$ — and flag in Section 6 that this is a title-level thesis, not one we can substantiate further from the supplied material. [12] The Two-Level Lie document, v2.0, adds Part II: the Approximation Entropy, described as a meta-mathematical framework quantifying the irreducible cost of translating continuous bosonic physics to discrete digital computation, introducing the Rosetta Constant, the Trotter Wall, and the Fractal Wall theorem with a falsifiable calorimetry experimental programme.

   from 2. Background and Related Work

Where the claim stops

1. The identification $\varepsilon = 1/Q^{2}$ uses worst-case Farey spacing; a typical-case treatment would change the scaling from $\log_{2}(1/\varepsilon)$ to roughly $\tfrac{1}{2}\log_{2}(1/\varepsilon)$, halving the bit cost — a factor-of-two ambiguity we cannot resolve from the supplied material.

   from 6. Discussion

2. If physical law is not representable by rationals — e.g. if genuinely irrational constants enter irreducibly at the level of observable quantities — the grid picture fails and $H_{\mathrm{app}}$ measures only our storage, not nature's cost.

   from 6. Discussion

3. The framework's own falsifiability hook is the calorimetry experimental programme of [12]: if the predicted irreducible cost of continuous-to-digital translation has no calorimetric signature, the physical (as opposed to informational) reading of approximation entropy is falsified.

   from 6. Discussion

Where to read

1. Abstract218 words · 1 min
2. 1. Introduction296 words · 1 min
3. 2. Background and Related Work905 words · 4 min
4. 3. Methods315 words · 1 min
5. 4. Analysis400 words · 2 min
6. 5. Results122 words · 1 min
7. 6. Discussion360 words · 2 min
8. 7. Conclusion119 words · 1 min